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The influence of theoretical mathematical foundations on teaching and learning: a case study of whole numbers in elementary school

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Abstract

This paper examines the existence and impact of theoretical mathematical foundations on the teaching and learning of whole numbers in elementary school in France. It shows that the study of the New Math reform –which was eventually itself replaced in the longer term – provides some keys to understanding the influence of mathematical theories on teaching and learning. The paper studies changes related to place value, a notion that was deeply impacted by the introduction of numeration bases other than ten in 1970, and their subsequent removal in the 1980s. What the author terms ‘numeration units’ (ones, tens, hundreds, thousands, etc.) and ‘powers-of-ten written in figures’ (1, 10, 100, 1000, etc.) are key tools for describing and understanding changes. The author identifies two theories that have formed the basis for place value teaching in the twentieth century, and examines some aspects of their influence. The paper also addresses epistemological issues in the relation between academic mathematics and school mathematics, and highlights the role of units in the teaching of basic arithmetic.

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Notes

  1. In France, the New Math elementary school syllabus was published in 1970. This was followed by a counter-reform, with syllabi published between 1977 and 1980 (depending on the class level).

  2. The wording is somewhat different: e.g., ‘scholarly knowledge’ corresponds to ‘mathematical theory’, while ‘knowledge to be taught’ corresponds to ‘the planned mathematics curriculum’.

  3. The early post-reform period saw an attempt to theorize based on ‘number words’ computations: e.g., 492 = (4 × 100) + (4 × 20) + 12, which reads four (4) hundreds (100), four (4) twenties (20), twelve (12) (in French).

  4. Unlike the English language, the French language does not use the number name to form the unit name. However, working in different bases requires using several names for the unit of a given order: e.g., twos, threes, tens in the first order; fours, nines, hundreds, in the second.

  5. Other units are taught in the context of continuous quantities such as length, mass, etc. In the context of numbers, the wording is restricted to the number 1 (and the first place in the numeration chart in French).

References

  • Artigue, M., & Robinet, J. (1982). Numération à l'école élémentaire [Numeration in elementary school]. Educational Studies in Mathematics, 13(2), 155–175.

  • Arzarello, F., & Bartolini-Bussi, M. G. (1998). Italian trends in research in mathematics education: A national case study in the international perspective. In J. Kilpatrick & A. Sierpinska (Eds.), Mathematics education as a research domain: A search for identity (Vol. 2, pp. 243–262). Dordrecht: Kluwer.

  • Barbé, J., Bosch, M., Espinoza, L., & Gascón, J. (2005). Didactic restrictions on the teacher's practice: The case of limits of functions in Spanish high schools. Educational Studies in Mathematics, 59, 235–268.

    Article  Google Scholar 

  • Bednarz, N., & Janvier, B. (1982). The understanding of numeration in primary school. Educational Studies in Mathematics, 13(1), 33–57.

    Article  Google Scholar 

  • Bergé, A. (2008). The completeness property of the set of real numbers in the transition from calculus to analysis. Educational Studies in Mathematics, 67(3), 217–235.

    Article  Google Scholar 

  • Bezout, E., & Reynaud, A. A. L. (1821). Traité d'arithmétique à l'usage de la marine et de l'artillerie [Treatise of arithmetic for marine and artillery] 9th edition.

  • Blanc, J.-P., Bramand, P., Debû, P., Gély, J., Peynichou, D., & Vargas, A. (2002). Pour comprendre les mathématiques. CE2 [To understanding mathematics. Grade 3]. Paris: Hachette.

  • Bosch, M., & Chevallard, Y. (1999). La sensibilité de l'activité mathématique aux ostensifs: Objet d'étude et problématique. [Sensitivity to ostensive objects in doing mathematics: Object of study and research problem]. Recherches en didactique des mathématiques, 19(1), 77–123.

  • Bosch, M., Gascón, J., & Trigueros, M. (2017). Dialogue between theories interpreted as research praxeologies: The case of APOS and the ATD. Educational Studies in Mathematics, 95(1), 39–52.

    Article  Google Scholar 

  • Boucheny, G., & Guérinet, A. (1931), L’arithmétique au cours élémentaire [Arithmetic in the 2nd and 3rd grades]. Paris: Larousse.

  • Bronner, A. (2008). La question du numérique dans l’enseignement du secondaire [The numeric issue in secondary teaching]. In A. Rouchier & I. Bloch (Eds.), Perspectives en didactique des mathématiques (pp. 17–45). Grenoble: La pensée sauvage.

  • Bruner, J. S. (1966). Toward a theory of instruction. Harvard: Harvard University Press.

    Google Scholar 

  • Castela, C., & Romo Vázquez, A. (2011). Des mathématiques à l'automatique: Étude des effets de transposition sur la transformée de Laplace dans la formation des ingénieurs [From mathematics to automation: Study of effects of transposition on Laplace transform in engineering education]. Recherches en didactique des mathématiques, 31(1), 79–130.

  • Chambris, C. (2008). Relations entre les grandeurs et les nombres dans les mathématiques de l'école primaire [Relations between quantities and numbers in mathematics for elementary school]. (Unpushed doctoral dissertation). Université Paris–Diderot, Paris.

  • Chambris, C. (2010). Relations entre grandeurs, nombres et opérations dans les mathématiques de l'école primaire au 20e siècle: Théories et écologie [Relations between quantities, numbers and operations in mathematics for elementary school in the 20th century: Theories and ecology]. Recherches en didactique des mathématiques, 30(3), 317–366.

  • Champeyrache, G., & Fatta, J.-C. (2002). Le nouveau Math élem. CE2 [New elementary math grade 3]. Paris: Belin

  • Chevallard, Y. (1985). La transposition didactique [Didactic transposition]. Grenoble: La pensée sauvage.

  • Chevallard, Y. (1997). Familière et problématique, la figure du professeur [The teacher as a colloquial and problematic figure]. Recherches en didactique des mathématiques, 17(3), 17–54.

  • Deblois, L. (1996). Une analyse conceptuelle de la numération de position au primaire [A conceptual analysis of numeration in elementary school]. Recherches en didactique des mathématiques, 16(1), 71–127.

  • Eiller, R., Brini, R., Martineu, M., Ravenel, S., & Ravenel, R. (1979). Math et calcul. CE2 [Math and computation. Grade 3]. Paris: Hachette.

  • Eiller, R., & Martineu, M. (1972). Math et calcul. CE2 [Math and computation. Grade 3]. Paris: Hachette.

  • Eiller, R., Martineu, M., Brini, R., Cornibé, R., & Pradillon, F. (1971). Math et calcul. CE1 [Math and computation. Grade 2]. Paris: Hachette.

  • ERMEL. (1978). Apprentissages mathématiques à l'école élémentaire. Cycle élémentaire. (Vol. 2) [Mathematical learning in elementary school. Second and third grades. Vol. 2]. Paris: SERMAP-Hatier.

  • Ernest, P. (2006). A semiotic perspective of mathematical activity: The case of number. Educational Studies in Mathematics, 61(1–2), 67–101.

    Article  Google Scholar 

  • Freudenthal, H. (1973). Mathematics as an educational task. Dordrecht: Reidel.

    Google Scholar 

  • Furinghetti, F., Menghini, M., Arzarello, F., & Giacardi, L. (2008). ICMI renaissance: The emergence of new issues in mathematics education. In M. Menghini, F. Furinghetti, L. Giacardi, & F. Arzarello (Eds.), The first century of the international commission on mathematical instruction (1908–2008). Reflecting and shaping the world of mathematics education (pp. 131–147). Rome: Istituto della Enciclopedia Italiana.

  • Fuson, K. C. (1990). Conceptual structures for multiunit numbers: Implications for learning and teaching multidigit addition, subtraction, and place value. Cognition and Instruction, 7(4), 343–403.

    Article  Google Scholar 

  • Gispert, H. (2010). Rénover l'enseignement des mathématiques, la dynamique internationale des années 1950 [Renewing mathematics teaching, the international dynamic of the 1950s]. In R. d'Enfert & P. Kahn (Eds.), En attendant la réforme. Politiques éducatives et disciplines scolaires sous la Quatrième République (pp. 131–143). Grenoble: PUG.

  • Griesel, H. (2007). Reform of the construction of the number system with reference to Gottlob Frege. ZDM, 39(1–2), 31–38.

    Article  Google Scholar 

  • Harlé, A. (1987). L'image du nombre dans les manuels d'arithmétique de l'enseignement primaire au début du XXème siècle [Image of numbers in arithmetic textbooks from early 20th century]. In Groupe d’Histoire des Mathématiques (Ed.), Fragments d'histoire des mathématiques II (pp. 22–84). Paris: APMEP.

  • Howe, R. (2015). The most important thing for your child to learn about arithmetic. In X. Sun, B. Kaur, & J. Novotná (Eds.), Proceedings of the twenty-third ICMI study: Primary mathematics study on whole numbers (pp. 107–114). China: University of Macao.

    Google Scholar 

  • ICMI. (2014). Discussion document of the twenty-third ICMI study: Primary mathematics study on whole numbers. Retrieved from http://www.mathunion.org/fileadmin/ICMI/docs/ICMIStudy23_DD.pdf

  • Kilpatrick, J. (2012). The new math as an international phenomenon. ZDM, 44(4), 563–571.

    Article  Google Scholar 

  • Kline, M. (1973). Why Johnny can't add: The failure of the new math. New York: St. Martin's Press.

    Google Scholar 

  • Lamon, S. J. (1996). The development of unitizing: Its role in children's partitioning strategies. Journal for Research in Mathematics Education, 27(2), 170–193.

    Article  Google Scholar 

  • Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers' understanding of fundamental mathematics in China and the United States. Mahwah: Lawrence Erlbaum Associates.

    Google Scholar 

  • Ma, L. (2013). A critique of the structure of U.S. elementary school mathematics. Notices of the AMs, 60(10), 1282–1296.

    Article  Google Scholar 

  • Menotti, G., & Ricco, G. (2007). Didactic practice and the construction of the personal relation of six-year-old pupils to an object of knowledge: Numeration. European Journal of Psychology of Education, 22(4), 477–495.

    Article  Google Scholar 

  • Moreira, P. C., & David, M. M. (2008). Academic mathematics and mathematical knowledge needed in school teaching practice: Some conflicting elements. Journal for Mathematics Teacher Education, 11(1), 23–40.

    Article  Google Scholar 

  • Mounier, E. (2013). Y a-t-il des marges de manœuvres pour piloter la classe durant une phase de bouclage? [Is there a degree of latitude for the teacher when he ends the lesson?]. Recherches en didactique des mathématiques, 33(1), 79–113.

  • Neyret, R. (1995). Contraintes et détermination des processus de formation des enseignants: Nombres décimaux, rationnels et réels dans les Instituts Universitaires de Formation des Maîtres [Constraints and determinations of teacher education: Decimals, rational and real numbers in training institutes] Thèse. Grenoble: Université Joseph Fourier Grenoble.

  • Otte, M. F. (2007). Mathematical history, philosophy and education. Educational Studies in Mathematics, 66(2), 243–255.

    Article  Google Scholar 

  • Otte, M. F. (2011). Evolution, learning, and semiotics from a Peircean point of view. Educational Studies in Mathematics, 77(2–3), 313–329.

    Article  Google Scholar 

  • Perret, J. F. (1985). Comprendre l'écriture des nombres [Understanding written numerals]. Bern: P. Lang.

  • Ross, S. H. (1989). Parts, wholes and place value: A developmental view. Arithmetic Teacher, 36(6), 47–51.

    Google Scholar 

  • Thanheiser, E. (2009). Preservice elementary school teachers' conceptions of multidigit whole numbers. Journal for Research in Mathematics Education, 40(3), 251–281.

    Google Scholar 

  • Thompson, I. (1999). Implications of research on mental calculations for the teaching of place value. Curriculum, 20(3), 185–191.

    Google Scholar 

  • Wittmann, E. (1975). Natural numbers and groupings. Educational Studies in Mathematics, 6(1), 53–75.

    Article  Google Scholar 

Download references

Acknowledgements

This article is based on a paper that was presented at the pre-conference for the 23rd ICMI study on Whole Number Arithmetic in Macau (China), in June 2015.

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  1. Christine Chambris

    Present address: Laboratoire de didactique André Revuz (EA 4434), UA, UCP, UPD, UPEC, URN, Université de Cergy-Pontoise, Cergy-Pontoise, France

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    Chambris, C. The influence of theoretical mathematical foundations on teaching and learning: a case study of whole numbers in elementary school. Educ Stud Math 97, 185–207 (2018). https://doi.org/10.1007/s10649-017-9790-3

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